By that logic, we could also use "he" and "she" interchangeably.
So do I, but my point is that "they/their" does agree with both singular and plural subjects, just as "you" does, for all the reasons I gave.
Plural subjects: Children, what are you doing? I can teach them to tie their shoelaces.
Singular subject: John, what are you doing? I can teach that child to tie their shoelaces.